Critical theory of the two-channel Anderson impurity model

Transcription

1 PHYSICAL REVIEW B 68, Critical theory of the two-channel Anderson impurity model Henrik Johannesson Institute of Theoretical Physics, Chalmers University of Technology and Göteborg University, SE Göteborg, Sweden N. Andrei and C. J. Bolech* Center for Materials Theory, Serin Physics Laboratory, Rutgers University, Piscataway, New Jersey , USA Received 20 December 2002; published 20 August 2003 We construct the boundary conformal field theory that describes the low-temperature behavior of the twochannel Anderson impurity model. The presence of an exactly marginal operator is shown to generate a line of stable fixed points parametrized by the charge valence n c of the impurity. We calculate the exact zerotemperature entropy and impurity thermodynamics along the fixed line. We also derive the critical exponents of the characteristic Fermi edge singularities caused by time-dependent hybridization between conduction electrons and impurity. Our results suggest that in the mixed-valent regime (n c 0, 1) the electrons participate in two competing processes, leading to frustrated screening of spin and channel degrees of freedom. By combining the boundary conformal field theory with the Bethe-Ansatz solution we obtain a complete description of the low-energy dynamics of the model. DOI: /PhysRevB I. INTRODUCTION PACS numbers: a, Hr, s In recent years, a growing class of materials has been shown to exhibit metallic behaviors that violate Landau s Fermi liquid theory. 1 Examples include the normal phases of underdoped and optimally doped high-t c cuprates, 2 a variety of quasi one-dimensional conductors ranging from single-wall carbon nanotubes 3 to the Bechgaard salts 4 several artificially designed nanostructures, 5 as well as certain cerium- and uranium-based heavy fermion alloys. 6 In light of the spectacular success of Landau s theory in explaining the properties of conventional metals, the proliferation of experimental systems that depart from its predictions presents a challenge to the theorist. Not forming a Fermi liquid, the mobile electrons of these systems cannot be adiabatically connected to a noninteracting electron gas, and their description requires a theoretical approach of a different brand. A particularly intriguing case is that of the heavy fermion compound UBe 13. Like several other metallic materials containing rare earth or actinide ions with partially filled f-electron shells UBe 13 exhibits manifest non-fermi-liquid behavior at low temperatures. In particular, the specific heat shows a TlnT behavior all the way down to the superconducting transition at roughly 1 K, 7 suggestive of a twochannel overscreened Kondo effect driven by the f- electrons of the uranium ions. There are two possible integer valence configurations for uranium embedded in a Be 13 host: a 6 magnetic spin doublet in the 5 f 3 configuration and a 3 electrical quadrupolar doublet in the 5 f 2 configuration, the two levels being separated by an energy. Arguing that the extremely weak magnetic field dependence of the material 6 excludes the usual magnetic Kondo effect, Cox 8 proposed that the observed anomalies instead derive from a quenching of the quadrupolar degrees of freedom by the local orbital motion of the 8 conduction electrons in the material with the electron spin, providing the two channels required for overscreening. The proposal has been controversial, 9 however, and experimental data on the nonlinear magnetic susceptibility indicate that the low-lying magnetic excitations are rather predominantly dipolar in character. 10 Whether the energy splitting between the two doublets is sufficiently large for a quadrupolar Kondo scenario to become viable is an additional unresolved issue. Aliev et al. 11 have suggested that the magnetic and quadrupolar doublets may in fact be near degenerate, leading instead to a mixed-valent state with a novel type of interplay between quadrupolar and magnetic Kondo-type screening. Indirect support for this hypothesis comes from nonlinear susceptibility 11 and other 12 measurements on the thoriated compound U 0.9 Th 0.1 Be 13. The presence of thorium ions suppresses the superconducting transition in the undoped material, and one observes instead a crossover from a magnetic to a quadrupolar ground state at low temperatures. The simplest model that captures a possible mixed-valent regime with both magnetic and quadrupolar character is the two-channel Anderson single-impurity model In this model the conduction electrons carry both spin, and quadrupolar quantum numbers and hybridize via a matrix element V with a local uranium ion. The ion is modeled by a quadrupolar (5 f 2 ) doublet created by a boson operator b and a magnetic (5 f 3 ) doublet created by a fermion operator f. Strong Coulomb repulsion implies that the localized f levels can carry at most one electron, and this condition is implemented as an operator identity for the pseudoparticles: f f b b 1, where denotes the conjugate representation. With these provisos the Hamiltonian is written as where HH bulk H ion H hybr, 1 H bulk dx: xi x x:, 2 H ion s f f q b b, /2003/687/ /$ The American Physical Society

2 HENRIK JOHANNESSON, N. ANDREI, AND C.J. BOLECH H hybr V 0b f f b 0. 4 Here the conduction electrons are described by onedimensional fields (x) matching the 8 representation appearing in the reduction of 3 6 and thus coupling to the impurity. The electron fields are chiral, obtained as usual 16 by reflecting the outgoing radial (x0) 8 component of the original three-dimensional problem to the negative x axis and disregarding components that do not couple to the ion at x0. The spectrum is linearized around the Fermi level, and the Fermi velocity is set to unity, with the resulting density of states being 1/(2). The normal ordering is taken with respect to the filled Fermi sea, and the energies s and q are those of the magnetic and quadrupolar doublets, respectively. Recently two of us presented an exact solution of the model, based on a Bethe-ansatz construction. 17 A complete determination of the energy spectrum and the thermodynamics was given, allowing a full description of the evolution of the impurity from its high-temperature behavior with all four impurity states being equally populated down to the lowenergy dynamics characterized by a line of fixed point Hamiltonians H*(,) where s q and V 2 ( is held fixed in what follows. It is found that the line of fixed points is characterized by 0 a zero-temperature entropy S imp k B ln2 and a specific heat C imp v TlnT typical of the two-channel Kondo fixed point. However, the physics along the line varies with. Consider n c f f, the amount of charge localized at the impurity. For, one finds that n c 1, signaling the magnetic integral valence regime. At intermediate temperatures a magnetic moment forms which undergoes frustrated screening as the temperature is lowered, leading to zero-temperature anomalous entropy and anomalous specific heat. For the system is in the quadrupolar integral regime and a quadrupolar moment forms. In the mixed-valence regime, similar low-temperature behavior is observed though without the intermediate regime of moment formation. In more detail, each point on the line of fixed points is characterized by two energy scales T l,h (). These scales describe the quenching of the entropy as the temperature is lowered: the first stage taking place at the high-temperature scale T h, quenching the entropy from k B ln4 to k B ln2, the second stage at T l, quenching it from k B ln2 to k B ln2. In the integral valence regime, the two scales are well separated and as long as the temperature falls between these values a moment is present magnetic or quadrupolar depending on the sign of ), manifested by a finite temperature plateau S imp k B ln2 in the entropy. It is quenched when the temperature is lowered below T l, with T l T K in this regime. For the scales are equal, T l ()T h (), and the quenching occurs in a single stage. In the present paper we shall concentrate on the lowenergy regime and give a detailed description of the line of fixed points in terms of boundary conformal field theory the low-energy effective Hamiltonian. It was argued by Affleck and Ludwig 18,19 that the low-energy regime of a quantum impurity problem is described by a boundary conformal field theory BCFT with a conformally invariant boundary condition replacing the dynamical impurity. However, to identify the effective low-energy theory characterizing a given microscopic impurity model, such as Hamiltonian 1, it is necessary to employ more powerful methods, valid over the full energy range, that can connect the microscopic Hamiltonian to the effective low-energy theory. One needs in principle to carry out a full renormalization group calculation or, when available, use an exact solution to extract the BCFT. Although applicable only in the neighborhood of a lowtemperature fixed point, a BCFT formulation has certain advantages. First, it provides an elegant scheme in which to analyze the approach to criticality. The leading scaling operators can be identified explicitly, yielding access to the low-temperature thermodynamics in closed analytical form. These can be matched with the expressions from the Betheansatz solution, allowing the determination of the parameters appearing in the BCFT. This will be carried out below. Secondly and more importantly having identified the scaling operators, the asymptotic dynamical properties Green s functions, resistivities, optical conductivities, etc. can in principle be calculated. These results will be presented in a subsequent work. The paper is organized as follows. In Sec. II we review the basics of BCFT, with particular focus on applications to quantum impurity problems. In Sec. III we derive the specific BCFT which describes the low-temperature physics of the two-channel Anderson model in Eq. 1. Some important features of this BCFT are discussed, and in this section we also apply it to the Fermi edge singularity problem for this model. In Sec. IV we then combine the results obtained with the Bethe-ansatz solution 17 to analytically calculate the zerotemperature impurity entropy, the impurity contribution to the low-temperature specific heat, and the linear and nonlinear impurity magnetic susceptibilities. Section V, finally, contains a summary and a discussion of our results. II. BOUNDARY CONFORMAL FIELD THEORY APPROACH The BCFT approach to quantum impurity problems is well reviewed in the literature, and we here only collect some basic results so as to fix conventions and notation. The key idea is to trade a local impurity-electron interaction for a scale invariant boundary condition on the critical theory that represents the extended electron degrees of freedom. In the presence of a boundary, the left- and rightmoving parts of the fields (t,x) in the critical theory are identified via analytic continuation beyond the boundary x 0, such that t,x L t,x R t,x L t,x L t,x. Thus, the boundary effectively turns the fields nonlocal. The consequence of this can be delineated via the operator product expansion L z L z* j PHYSICAL REVIEW B 68, g j ( O j) zz* 2 L 0,, j z z*

3 CRITICAL THEORY OF THE TWO-CHANNEL ANDERSON... We have here introduced a complex variable zix, with the Euclidean time and x the spatial coordinate. The function g j (0 or 1) selects the boundary operators O j) L asso- ( ciated with the given boundary condition, j () being the ( dimension of O j) L ( L ). It follows from Eqs. 5 and 6 that an n-point function of a field close to the boundary turns into a linear combination of n-point functions of the chiral ( boundary operators O j) L. In particular, this implies that the boundary scaling dimension bound of that governs its autocorrelation function close to the boundary, t,x0,xt,x0,xt 2 bound, tx, is precisely given by the scaling dimension of the leading boundary operator appearing in Eq. 6. This sets the strategy for treating a quantum impurity problem: identify first the particular boundary condition that plays the role of the impurity interaction. Given that this boundary condition is indeed scale invariant together with the original bulk theory, one can then use the machinery of BCFT to extract the corresponding boundary scaling dimensions. As these determine the asymptotic autocorrelation functions cf. Eq. 7 the finite-temperature properties due to the presence of the boundary alias the impurity are easily accessed from standard finite-size scaling by treating the Euclidean time as an inverse temperature. To identify the right boundary condition it is convenient to exploit a well-known result 23 from conformal field theory, relating the energy levels in a finite geometry to the boundary scaling dimensions of operators in the semi- infinite plane. More precisely, consider a conformally invariant theory defined on the strip wuiv u, 0 v with u the Euclidean time and v the space coordinate. Then impose a conformally invariant boundary condition, call it A, at the edges v0 and v, and map the strip onto the semi-infinite plane zixx0, using the conformal transformation z exp (w/l) implying boundary condition A at x0). With E 0 the ground-state energy, one has 7 E n E 0 n, 8 where E n is the spectrum of excited energy levels in 0 v and n is the spectrum of boundary scaling dimensions in the semi-infinite plane. The problem is thus reduced to determining the finite-size spectrum of the theory. In certain privileged cases this can be done by direct calculation. Alternatively, one hypothesizes a boundary condition and compares the consequences with known answers. An example is the fusion hypothesis of Affleck and Ludwig. 19 One here starts with some known, trivial boundary condition on the critical theory, with no coupling to the impurity. For free electrons carrying charge, spin and maybe some additional flavor degrees of freedom the spectrum organizes into a conformal tower structure, 24 with the charge, spin, and flavor sectors glued together so as to correctly represent the electrons (Fermi liquid gluing condi- PHYSICAL REVIEW B 68, tion). Then, turning on the electron-impurity interaction, its only effect according to the fusion hypothesis is to replace this gluing condition by some new nontrivial gluing of the conformal towers. In the case of the m-channel Kondo problem, the new gluing condition is obtained via Kac- Moody fusion 25 with the spin-1/2 primary operator that corresponds to the Kondo impurity. As a consequence, the conformal tower with spin quantum number j s is mapped onto new towers labeled by j s, where j s j s 1 2, j s 1 2 1,..., min j s 1 2,m j s 1 2. This is the essence of the BCFT approach, as applied to the Kondo problem. For the two-channel Anderson model in Eq. 1 it is a priori less obvious which operator to use for fusion. Here we shall instead identify the finite-size spectrum by studying certain known limiting cases, guided by the exact Betheansatz solution of the model. 17 As it turns out, the spectrum thus obtained can be reconstructed by fusion with the leading flavor boundary operator representing the channel, or quadrupolar, degrees of freedom, but in addition there occurs an effective renormalization of the charge sector. This signals the novel aspect of the present problem. A technical remark may here be appropriate: Since the boundary scaling dimensions in Eq. 8 are connected to energy levels in a strip with the same boundary condition at the two edges, we are effectively considering a finite-size energy spectrum with two quantum impurities present, one at each edge of the strip. Formally, this can be taken care of by performing Kac- Moody fusion twice with the relevant primary operator (double fusion). This is an important point, not always fully appreciated in the literature. Let us finally mention that there exists another, more fundamental description of BCFT, 26 based on the notion of boundary states, particularly useful when studying zerotemperature properties of a quantum impurity problem. We shall give a brief exposition of it in Sec. IV.A, where we use it for analyzing the zero-temperature entropy contributed by the impurity. III. FINITE-SIZE SPECTRUM AND SCALING OPERATORS As we discussed in the previous section, the scaling dimensions of the boundary operators that govern the critical behavior are in one-to-one correspondence with the levels of the finite-size spectrum of the model. In the present case the spectrum can in principle be constructed from the exact Bethe-ansatz solution in Ref. 17, but this requires an elaborate analysis. The reason is that the solution takes the form of a so-called string solution even in the ground state. Since a string solution is valid only in the thermodynamic limit, an application of standard finite-size techniques 27 would lead to incorrect results. One may instead pursue another strategy and identify the finite-size spectrum hence the dimensions of the scaling operators that reproduces the results of the exact solution in the low-energy limit. In our case the search for the wanted spectrum can be carried out efficiently by combining symmetry arguments with known results for the finite-size spectra for two limiting cases of the Hamiltonian 1: the ordinary Anderson model and the two-channel Kondo model. We shall see that the low-energy spectrum

4 HENRIK JOHANNESSON, N. ANDREI, AND C.J. BOLECH thus obtained, Eq. 22 below, leads indeed to a line of fixed points characterized by entropy Sk B ln2 and susceptibility C V TlnT as required by the Bethe-ansatz solution. The fitting of the proposed spectrum to the solution also allows the determination of all thermodynamically relevant parameters in the BCFT. A. Identifying the critical theory Let us begin by reviewing the single-channel Anderson and Kondo models. The latter is obtained in the integervalence limit n c 1 with n c f f measuring the charge localized at the impurity site where a magnetic moment forms, signaling the entrance to the Kondo regime. 28 It is instructive to consider this case in some detail before proceeding to an analysis of the full model in Eq. 1. As is well known, the ordinary Anderson model 29 with integer valence n c 1 can be mapped onto the single-channel Kondo model via a Schrieffer-Wolff transformation. 28 To O(1/), with the length of the system, the finite-size energy spectrum of the Kondo model takes the form 19 EE Q2 N c j s j s 1 N 3 s, where the Fermi velocity has been set to unity, as in Eq. 1. Here QZ and j s 0,1/2 are charge and spin quantum numbers defined with respect to the filled Fermi sea and labeling the corresponding Kac-Moody primary states. 25 The positive integers N c and N s index the descendant levels in the associated U1 and SU(2) 1 conformal towers. The quantum numbers are constrained to appear in the combinations Q0 mod2, j s 1 2, 9 Q1 mod2, j s 0, 10 which define the Kondo gluing conditions for charge and spin conformal towers. We can rewrite this result in a form immediately generalizable to the Anderson model, viewing it as the n c 1 limit of the Anderson model spectrum. Redefine the charge quantum number in Eqs. 9 and 10: Q Q1. The offset mirrors the fact that the Anderson impurity carries charge in contrast to a Kondo impurity and, therefore, shifts the net amount of charge in the ground state compared to the Kondo case. We thus write for the finite-size spectrum of the Anderson model in the n c 1 limit: EE Q c N j s j s 1 N 3 s, 11 with Note that the constraints in Eqs. 12 are the same as those for the charge and spin conformal towers of free electrons (Fermi liquid gluing conditions). Also note that we have renormalized the ground-state energy, so that E 0 E(Q 0), by subtracting off a constant /4. The redefinition of charge quantum numbers implies a relabeling of levels: the Q level in Eq. 9 constrained by Eqs. 10 corresponds to the (Q1) level in Eq. 11 with the constraint 12, modulo the /4 shift. Let us now consider the spectrum of the Anderson model away from the Kondo limit, n c 1. Assuming the absence of any relevant operator in the renormalization group RG sense that could take the model to a new boundary fixed point and hence change the gluing conditions in Eqs. 12 we expect that the finite-size spectrum still has a Kac- Moody structure with Fermi liquid gluing conditions. Since the ground state now has an average surplus charge n c 1 compared to the case with an empty localized level, we conclude from Eq. 18 that EE Qn c 2 j s j s 1 N 3 s, n 2 c 2 N c 13 with Q and j s constrained by Eqs. 12. Note that this result has been extracted from the Kondo finite-size spectrum in Eqs. 9 and 10 simply by keeping track on the offset of charge with respect to the filled Fermi sea as the impurity acquires a nonzero-charge valence n c. The additional assumption that no relevant operator is generated as we move away from the Kondo limit n c 1 and that therefore the gluing conditions in Eqs. 10 remain unchanged will be shown to be self-consistent when we analyze the full twochannel Anderson model. We can corroborate our procedure by comparing our result in Eq. 13 with that obtained by Fujimoto and collaborators. 30 Applying standard finite-size techniques 27 to the exact Bethe-ansatz solution 31 of the Anderson model these authors found that PHYSICAL REVIEW B 68, EE Q 2 F j s j s 1 N 3 s O 2 1, 2 F N c 14 with Fermi liquid gluing 12 of Q and j s. The phase shift F ( F ) appearing in Eq. 14 is that of an electron with spin at the Fermi level, related to the average charge n c f f at the impurity site by the Friedel-Langreth sum rule 32 Q0 mod2, j s 0, F 2 n c. 15 Q1 mod2, j s Inserting Eq. 15 into Eq. 14, we immediately recover our result

5 CRITICAL THEORY OF THE TWO-CHANNEL ANDERSON... Before returning to the full two-channel Anderson model, let us point out that the duality between the charge shift Q Q1 in the integer valence limit n c 1 and the change of gluing conditions simply reflects the wellknown fact that the ordinary Kondo effect can formally be described as a local potential scattering of otherwise free electrons, causing a phase shift F /2 of their wave functions cf. Eq. 15. It is instructive to make this duality transparent by identifying the effective low-energy Hamiltonian H charge that produces the charge spectrum in Eq. 11. Introducing the charge currents Jx: x x: 16 where as before the normal ordering is taken with respect to the filled Fermi sea, one immediately recognizes the charge part of Eq. 11 as the spectrum of the Sugawara Hamiltonian 25 PHYSICAL REVIEW B 68, With this we recover the form of the conformal spectrum as derived by Fujimoto and Kawakami 34 from the exact solution. 35,36 By mapping the nontrivial Z 2 part of the twochannel Kondo scattering onto that of a restricted solid-onsolid model 37 coupled to the impurity, these authors elegantly circumvented the string solution problem mentioned above. In exact analogy with the single-channel case detailed above, we can match the finite-size two-channel Kondo spectrum defined by Eqs. 18 and 19 to that for the n c 1 limit of the two-channel Anderson model by performing a shift Q Q1 in Eqs. 18 and 19. The gluing conditions are accordingly modified and now take the form Q0 mod 2, j s 0 or 1, j f 1 2, H charge 1 4 dx:jxjx: 1 2 J0. 17 Q1 mod 2, j s 1 2, j f0 or The term J(0) explicitly reveals the presence of an effective local scattering potential. By redefining the charge quantum numbers in the integer valence limit, Q1 Q, this potential is disguised as a renormalized boundary condition, encoded in the Kondo gluing conditions 10. With these preliminaries let us now go back to the full two-channel Anderson model in Eq. 1, first considering the case when s q. From the Bethe-ansatz solution 17 one finds that in this limit the physics is that of the overscreened two-channel magnetic Kondo model. 33 This is expected, since for this case the ion has integer valence, with an associated magnetic moment of spin 1/2. The finitesize spectrum of the two-channel Kondo model has been derived from that of free electrons carrying spin and flavor via Kac-Moody fusion 25 with the spin-1/2 conformal tower. 19 One finds, to O(1/), EE Q2 N c j f j f 1 N 4 f. j s j s 1 N 4 s 18 Here QZ are U1 charge quantum numbers, while j s( f ) 0, 1/2, 1 are quantum numbers for the level-2 SU2 spin flavor primary states, with N c, N s, and N f labeling the corresponding descendant levels. The quantum numbers are restricted to appear in the combinations Q0 mod 2, j s 1 2, j f0 or 1, Q1 mod 2, j s 0 or 1, j f 1 2, 19 which define the two-channel Kondo gluing conditions 19 for the conformal towers. Since the charge quantum numbers in Eqs. 19 are defined modulo 2, we can make a shift Q Q2 in Eq. 18 without affecting the gluing conditions. The gluing conditions 20 can formally be obtained by starting with the gluing conditions for free electrons carrying spin and flavor: Q0 mod 2, j s 0, j f 0 or j s 1, j f 1, Q1 mod 2, j s 1 2, j f 1 2, 21 and then performing Kac-Moody fusion with the j f 1/2 flavor conformal tower: j f 0 1/2, j f 1/2 0, 1, j f 1 1/2. Alternatively, we can obtain Eqs. 20 from Eqs. 21 by fusion with the j s 1/2 spin conformal tower: j s 0 1/2, j s 1/2 0, 1, j s 1 1/2, concurrent with fusion with the Q1 charge conformal tower: Q Q1. In Sec. III B we shall see how these two equivalent schemes can be given a simple physical interpretation, considering the particular structure of the hybridization interaction in Eq. 4. It is important to realize that in contrast to the single-channel case, the two-channel Kondo gluing conditions in Eq. 19 cannot be traded for a local scattering potential by the charge shift Q Q1. Although a local scattering potential is generated cf. Eq. 23 below, the charge-renormalized gluing conditions 20 are not those of free electrons. A twochannel Kondo system forms a non-fermi liquid with properties very different from those of phase-shifted free electrons. Let us now turn to the case of noninteger impurity valence n c 1. As supplementary input we use the fact that the lowlying excited states of the model split into separate gapless charge, spin, and flavor excitations for any value of and hence n c ), as revealed by the exact Bethe-ansatz solution. 17 Since there is no change of symmetry of the model as we tune q and s away from the magnetic two-channel Kondo limit, one expects that a spectral U1 charge, SU(2) 2 spin, and SU(2) 2 flavor Kac-Moody structure is retained throughout the full range of magnetic and quadrupolar energies. This implies that in the absence of any relevant

6 HENRIK JOHANNESSON, N. ANDREI, AND C.J. BOLECH operators that could push the model to a fixed point of a different Kac-Moody structure, the finite-size spectrum must take the form EE Qn c n c 2 N c j s j s 1 4 N s j f j f 1 N 4 f, 22 with the values of the charge, spin, and flavor quantum numbers constrained by Eqs. 20. Note that the charge levels have again been shifted by a constant so that Q measures the number of conduction electrons added to or removed from the ground state i.e., E 0 E(Q0)]. Similar to the ordinary Anderson model, the charge spectrum in Eq. 22 implies the presence of an effective local scattering potential V e f f n c J The charge current J has scaling dimension 1, making V e f f in Eq. 23 into an exactly marginal boundary operator with the scaling field sampling the impurity charge valence n c. Provided that no relevant operators intervene under renormalization, it produces a line of stable fixed points parametrized by the value of n c or, equivalently,. This is in accordance with the Bethe-ansatz solution, 17 which established a line of low-temperature fixed points analytically connected to the overscreened two-channel magnetic Kondo fixed point at. Given the finite-size spectrum defined by Eqs. 20 and 22 we shall next identify the leading operator content of the scaling theory, following the double-fusion prescription outlined in Sec. II. In particular, we must check that no relevant operators are generated as we vary and move away from the integer valence limit n c 1. Performing a second Kac-Moody fusion 20 with the j f 1/2 flavor conformal tower, the gluing conditions 20 change to Q0 mod 2, j s 0 or 1, j f 0 or 1, Q1 mod 2, j s 1 2, j f PHYSICAL REVIEW B 68, These new gluing conditions ensure that the boundary condition in the half-plane geometry is time independent corresponding to having attached an Anderson impurity to each edge, x0 and x, of the strip; cf. Sec. II. As we are now effectively considering a finite-size spectrum with two quantum impurities present, the charge quantum number gets renormalized twice, with Q Qn c (x0)q from the impurity at the x0 edge and Q Qn c (x)q n c (x0)q from the x edge. Thus, the modifications of Q caused by n c at each edge of the strip cancel each other, and the double-fusion spectrum becomes independent of the impurity valence. The reason for the cancellation is the same as in the single-channel case, where the sum rule 15 connects n c to the scattering phase shift caused by the impurity. Since the phase shifts at x0 and x have opposite signs at one edge a left-moving electron is reflected, while at the other edge a right-moving electron gets reflected, one must formally assign charge valences of opposite signs to the auxiliary impurities attached to x0 and x, respectively. We clearly expect the same situation to apply also in the twochannel case. By comparison with Eqs. 8 and 18 we thus read off for the possible boundary scaling dimensions c s f : c 1 8 Q2 N c, s 1 4 j s j s 1N s, f 1 4 j f j f 1N f, 25 with Q, j s, and j f constrained by Eqs. 24, and with N c, N s, N f positive integers. The gluing conditions in Eqs. 24 are identical to those for the spectrum of boundary scaling dimensions in the twochannel Kondo model. 19 Here we obtained them by double fusion with the j f 1/2 flavor conformal tower, while in the Kondo case they follow from double fusion with the spin-1/2 conformal tower. The equivalence of the two procedures is consistent with the Q mod 2 invariance of Eqs. 24. Double fusion with j f 1/2 in the empty valence limit turns into double fusion with j s 1/2 in the Kondo limit with a concurrent charge renormalization Q Q2. We shall elaborate on this point in the next section. The symmetries of the Hamiltonian in Eq. 1 impose further restrictions on the allowed Kac-Moody charge, spin, and flavor quantum numbers. Charge conservation implies that Q0 in Eqs. 25 and hence only the trivial identity conformal tower in the charge sector contributes to the spectrum of boundary scaling dimensions. Moreover, since the charge valence n c does not appear in Eqs. 25, no scaling dimension can depend on it. No relevant operator can therefore appear as n c is tuned away from the magnetic twochannel Kondo limit. This guarantees that the finite-size spectrum defined by Eqs. 24 and 25 remains valid throughout the full range of charge valence n c (0,1), with a line of stable fixed points parametrized by n c. Recall that this line is produced by the marginal operator V e f f in Eq. 23, associated with the single-fusion spectrum, Eqs. 20 and 22, which carries an explicit dependence on n c. Here V e f f plays the role of a boundary changing operator. 44 For a given value of n c it determines the locus on the fixed line to which the theory flows under renormalization. It may be tempting to conclude that the result obtained from double fusion, Eqs. 24 and 25, implies that the time-independent critical behavior is the same along the critical line, being insensitive to n c. This is correct with regard to critical exponents, but as we shall see below, amplitudes of various scaling operators pick up a dependence on n c, giving rise to very different physics as we move along the fixed line. The leading boundary operator contributed by the charge sector is the exactly marginal charge current J(0) : (0) (0):. It is formally identified as the first Kac

7 CRITICAL THEORY OF THE TWO-CHANNEL ANDERSON... Moody descendant 25 J(0)J 1 1 (c) (0) of the identity operator 1 (c) which generates the Q0 conformal tower. As we have seen, J(0) being marginal, it shows up manifestly in the low-energy spectral-generating Hamiltonian H charge in Eq. 17 and is responsible for producing the line of fixed points. The appearance of the charge current breaks particlehole symmetry, since J(0) J(0) under charge conjugation (x) (x). It is interesting to note that while the same symmetry is broken by the ionic pseudoparticle term H ion in Eq. 1 under f f, the pieces of the Hamiltonian involving the conduction electron field (x) H bulk and H hybr actually respect charge conjugation. This symmetry gets broken dynamically via the coupling to the pseudoparticles, allowing the charge current in Eq. 23 to enter the stage. 38 The next-leading boundary operator from the charge sector is that of the energy-momentum tensor, T (c) (0) :J(0)J(0):/4, appearing as the second Virasoro descendant 25 of the identity operator, T c (0)L 2 1 (c) (0). Although it carries scaling dimension 2 and hence is subleading to the charge current J(0), being a Virasoro descendant of the identity operator it has a nonvanishing expectation value and turns out to produce the same scaling in temperature as J(0). Although not central to the present problem, we shall briefly return to this question in Sec. IV. Focusing now on the spin and flavor sectors, conservation of total spin and flavor quantum numbers requires that all spin and flavor boundary operators must transform as singlets. The leading spin boundary operator with this property, O (s) (0) call it, has dimension s 3/2 and is obtained by contracting the spin-1 field (s) (0) which generates the j s 1 conformal tower with the vector of SU(2) 2 raising operators J (s) 1 :O (s) (0)J (s) 1 (s) (0). As expected, this is the same operator that drives the critical behavior in the twochannel magnetic Kondo problem. 19 The next-leading spin boundary operator is the energy momentum tensor T (s) (0) :J (s) (0) J (s) (0):/4 L 2 1 (s) (0), of dimension s 2. The two leading flavor boundary operators are obtained in exact analogy with the spin case. In obvious notation O ( f ) ( f (0)J ) 1 ( f ) (0) of dimension f 3/2, and T ( f ) (0) :J ( f ) (0) J ( f ) (0): L 2 1 ( f ) (0) of dimension f 2. Boundary operators with integer scaling dimensions generate an analytic temperature dependence of the impurity thermodynamics, 20 subleading to that coming from the two leading irrelevant operators O (s) (0) and O ( f ) (0). Hence, in what follows we shall focus on these latter operators. In the case of the two-channel Kondo problem the flavor operator O ( f ) (0) was argued to be effectively suppressed, at least for the case when the bare Kondo coupling is sufficiently small. 19 On dimensional grounds one expects that the spin scaling field s multiplying O (s) (0) in the scaling Hamiltonian is O(1/T K ), where T K is the Kondo scale for the crossover from weak coupling high-temperature phase to strong renormalized coupling low-temperature phase. The flavor operator, on the other hand, does not see this scale since the infrared divergences in perturbation theory which signal the appearance of a dynamically generated scale occur only in the spin sector. The only remaining scale is that PHYSICAL REVIEW B 68, of the bandwidth D which plays the role of an ultraviolet cutoff, implying that the flavor scaling field f multiplying O ( f ) (0)] is O(1/D). For a small bare Kondo coupling, T K D exp (1/)D, and the critical behavior is effectively determined by O (s) (0) alone. As we will show, the picture changes dramatically for the two-channel Anderson model. The Bethe-ansatz solution 17 shows that there are two dynamically generated temperature scales T () present in this model. They determine and parametrize the thermodynamic response of the system. Consider for example the quenching of the entropy as the temperature is lowered: in the magnetic moment regime ( ) the two scales are widely separated, T T. For temperatures in the range T TT charge and flavor fluctuations are suppressed and the entropy shows a plateau corresponding to the formation of a local moment. The moment undergoes frustrated screening when the temperature is lowered below T, reaching zero-point entropy Sk B ln2, and n c 1. As decreases T and T approach each other and the range over which the magnetic moment exists decreases too. It disappears i.e., T T ) when. At this point no magnetic moment forms. However, the system still flows to a frustrated two-channel Kondo fixed point, with entropy Sk B ln2, but with n c 1/2. Continuing along the critical line, the two scales trade places, and eventually, at the quadrupolar critical endpoint (), one finds that T T. Since the irrelevant spin flavor operator is expected to dominate the critical behavior at the magnetic quadrupolar fixed point, we are led to conjecture that the corresponding scaling field is parametrized precisely by 1/T (1/T ). With this scenario played out, the relative importance of the two boundary operators changes continuously as one moves along the fixed line, with each operator ruling at its respective critical end point. In the next section we shall prove our conjecture by computing the impurity specific heat and susceptibility following from the scaling Hamiltonian and comparing it to the lowtemperature thermodynamics from the exact Bethe-ansatz solution. 7 We shall also use this comparison to fit the parametrization of the scaling fields. This fixes the critical theory completely. B. Application: Fermi edge singularities Before taking on this task, however, we shall exploit the BCFT scheme developed above to determine the scaling dimensions of the pseudoparticles which enter the Hamiltonian in Eqs. 3 and 4. The analogous problem for the ordinary Anderson model has been extensively studied, in part because of its close connection to the x-ray absorption problem. One is here interested in the situation where x-ray absorption knocks an electron from a filled inner shell of an ion in a metal into the conduction band. As an effect the conduction electrons experience a transient local potential at the ion which lost the core electron, and this typically produces a singularity in the x-ray spectrum ( F ) close to the Fermi level threshold F. Nozieres and de Dominicis, 42 using a simple model, showed that the exponent depends

8 HENRIK JOHANNESSON, N. ANDREI, AND C.J. BOLECH only on the phase shift which describes the scattering of conduction electrons from the core hole created by the x-ray. Fermi edge singularities linked to local dynamic perturbations are fairly generic, 43 and it is therefore interesting to explore how they may appear in a slightly more complex situation where a localized deep-hole type perturbation involves additional degrees of freedom. Indeed, the twochannel Anderson model in Eq. 1 provides an ideal setting for this as it accommodates a local quantum impurity carrying both spin and channel quadrupolar degrees of freedom. To set the stage, let us make a Gedankenexperiment and imagine that we suddenly replace a thorium ion in the host metal say, U 0.9 Th 0.1 Be 13 ) by a uranium ion, the sole effect being to introduce a localized quadrupolar or magnetic moment. To describe the response of the host to this perturbation, we consider the pseudoparticle propagators G 0e iht f e iht f 0, G 0e iht b e iht b with no summation over,. At time t0 the ground state 0 of the metal is that of free conduction electrons. At t 0 a pseudoparticle creating the localized moment is inserted into the metal, and the system evolves in time governed by the two-channel Anderson Hamiltonian in Eq. 1. At some later time t the pseudoparticle is removed, and one measures the overlap between the state obtained with the initial unperturbed ground state. It is important to realize that by inserting removing a pseudoparticle operator, one simultaneously turns on off the impurity terms in Eqs. 3 and 4 and, hence, changes the dynamics of the conduction electrons. As we reviewed in Sec. II, at low temperatures and close to a critical point, this corresponds to a change of the boundary condition which emulates the presence absence of the impurity. Hence, the pseudoparticle operators f and b are boundary changing operators, 26 and we can use the BCFT machinery to calculate their propagators in Eqs. 26 and 27. Our discussion closely follows that in Ref. 44. We start by considering the free-electron theory defined on the complex upper half-plane C zix;x0 with a trivial Fermi liquid boundary condition, call it A, imposed at the real axis x0. We denote by A;0 the ground state for this configuration. By injecting a pseudoparticle at time 0, the boundary condition for 0 changes to B, here labeling the boundary condition which corresponds to the nontrivial gluing condition in Eqs. 20 for the spectrum of the two-channel Anderson model. By mapping the half-plane to a strip wuiv;0 via the conformal transformation w(/2)lnz, the boundary of the strip at v also turns into type B after insertion of the pseudoparticle operator. Under the same transformation the pseudoparticle propagator in C, A;0O i ( 1 )O i ( 2 )A;0( 1 2 ) 2x i with x i the dimension of O i,i f or b) transforms as PHYSICAL REVIEW B 68, x i 2 sinh 2x 2 u 1u 2 i n AA;0O i 0AB;n 2 e (E AB AA n E0 )(u2 u 1 ), 28 where on the right-hand side of Eq. 28 we have inserted a complete set of states AB;n,n0,1,... of energies E n AB, defined on the strip with boundary condition A (B) at u 0 (u). In the limit u 2 u 1, 2 sinh 2 u 1u 2 2xi e[x i (u 2 u 1 )/]. 29 It follows, by comparison with Eq. 28, that x i E n AB E AA i 0, 30 where n i is the lowest energy state with a nonzero matrix element in the sum in Eq. 28. The pseudoparticle spectrum I i () close to the Fermi level is obtained by Fourier transforming the corresponding propagator in Eq. 26 or 27, and it follows that 1 I i, i f, b, 31 F 12x i with a singularity when x i 1/2. Let us consider first the slave boson b, carrying quantum numbers Q0, j s 0, j f 1/2. Its scaling dimension x b is given by Eq. 30 where E AB ni is obtained from Eq. 22 by putting Q0, j s 0, j f 1/2, and with E AA 0 0 with proper normalization of energies. To be able to compare energies corresponding to different boundary conditions it is important to keep the overall energy normalization fixed, and hence we must remove the normalization constant n 2 c /8 in Eq. 22 before reading off the answer. Recall that this normalization constant is expressly designed to remove the Q0 contribution from the charge sector to the ground-state energy. We thus obtain x b n c Similarly, we obtain for the pseudofermion (Q1, j s 1/2, j f 0) x f n c 1 8 n c It is interesting to compare Eqs. 32 and 33 with the corresponding scaling dimensions in the ordinary Anderson model, obtained in Refs. 39 and 41: x b n c 2 /2, x f 1/2 n c /2n c 2 /4. We conclude that by opening an additional channel for the electrons, the singularity in the slave boson spectrum in Eq. 31 corresponding to x-ray photoemission gets softer, whereas for the pseudofermion spectrum corre

9 CRITICAL THEORY OF THE TWO-CHANNEL ANDERSON... sponding to x-ray absorption the opposite is true. One can show that this trend is systematic and gets more pronounced as the number of channels increases. 45 We also note that the values of the pseudoparticle scaling dimensions as obtained by various approximation schemes, e.g., the non-crossing approximation NCA and large-n calculations, 46 are not in agreement with the exact results presented here. Before closing this section, let us return to the Gedankenexperiment where we inserted an empty degenerate impurity level by adding a slave boson to the filled Fermi sea. The slave boson carries only flavor j f 1/2, and it follows that the allowed values of the quantum numbers of the combined system are obtained by coupling j f 1/2 to the flavor quantum numbers j f of the conduction electrons leaving charge and spin untouched. Since j f and j f label conformal towers, this coupling is precisely governed by conformal fusion with j f 1/2: j f j1/2, f j1/21, f..., min( j f 1/2, 3/2 j). f By letting the system relax, a nonzerocharge valence n c may then build up at the impurity site as determined by the particular value of s q ). Formally, this process is captured by the charge renormalization Q Qn c in the finite size-spectrum 22, taking place concurrently with the fusion in flavor space which produces the nontrivial gluing conditions 20. Running the Gedankenexperiment with a pseudofermion instead carrying charge and spin, but no flavor leads to a scenario dual to the one above, where the coupling is now governed by fusion with j s 1/2 in the spin sector and Q1 in the charge sector cf. the discussion after Eq. 21. As the system relaxes an average charge n c goes to the impurity site, again leading to the renormalization Q Qn c in the spectrum 22. We conclude that our key result, Eqs. 20 and 22 obtained in Sec II A via a formal analysis can be given a natural interpretation by considering the pseudoparticle structure of the hybridization interaction in Eq. 4. IV. LOW-TEMPERATURE THERMODYNAMICS A. Zero-temperature entropy The Bethe-ansatz solution 17 of the model reveals that the line of fixed points is characterized by a zero-temperature entropy S imp k B ln2 typical of the two-channel Kondo fixed point. As we shall show next, this property emerges naturally when feeding in our result from Sec. II A into the general BCFT formalism developed by Cardy. 26 Recall that the key ingredient in this formalism is the modular invariance 24 of a conformal theory. Applied to a model defined on a cylinder of circumference 1/T and length, and with conformally invariant boundary conditions A and B imposed at the open ends, this means that its partition function Z AB is invariant under exchange of space and Euclidean time variables. In our case, B is the boundary condition obtained from A by attaching an Anderson impurity to one of the edges of the cylinder with A a trivial Fermi liquid boundary condition, to be defined below. Now, let H* be the critical Hamiltonian that corresponds to the spectrum given by Eqs. 20 and 22. Then H AB (/)H* generates time translations around the cylinder specified above. The partition function Z AB can hence be written Z AB Tr e H AB a n a AB a e /. 34 Here a is a character of the conformal U1SU(2) 2 SU(2) 2 algebra, with a(q, j s, j f ) labeling a product of charge, spin, and flavor conformal towers constrained by a Eqs. 21 and with n AB counting its degeneracy within the spectrum of H AB. By a modular transformation the role of space and time is exchanged, and the partition function now gets expressed as Z AB Ae H B, 35 where H is the transformed Hamiltonian generating translations along the cylinder, in the space direction, with A and B boundary states 26 implicitly defined by Eq. 35. By inserting a complete set of Ishibashi states a m a;m L a;m R into Eq. 35 with m running over all states in an L/R product of charge, spin, and flavor conformal towers indexed by a), one obtains 24 Z AB a AaaB a e 4/. 36 Equating Eqs. 34 and 36 one can immediately read off Cardy s equation 26 b n b AB S ab AaaB, 37 where S ab is the modular S matrix defined by a e / b S ab b e 4/, 38 with a and b indexing products of charge, spin, and flavor conformal towers. It follows that the nontrivial boundary state B which emulates the presence of the impurity is related to A via the identity 26 abaa S ad S a0, 39 with 0 labeling the product of identity conformal towers and d the product of conformal towers with which the Kac- Moody fusion of the finite-size spectrum is performed. As we found in Sec. II A, for the two-channel Anderson model, d(q0, j s 0, j f 1/2). Factoring the modular S matrices in charge, spin, and flavor, with a(q, j s, j f ) constrained by Eqs. 21 but otherwise arbitrary, one obtains S ad S a0 PHYSICAL REVIEW B 68, c S Q0 c S Q0 s S js 0 f S j f 1/2 s f S js 0S j f 0 f S j f 1/2 f S j f The modular S matrix for the SU(2) 2 flavor symmetry is given by

10 HENRIK JOHANNESSON, N. ANDREI, AND C.J. BOLECH S j f j f 1 2 sin 2 j f12 j f 1 4, 41 and it follows from Eqs. 39 and 40 that sin 2 f1 2 j abaa sin 4 f1. 2 j 42 We now have the tools for deriving the impurity entropy. Following the analogous analysis of the multichannel Kondo model in Ref. 47 we take the limit / in Eq. 36 i.e., we take after the large-volume limit ). As a result, only the character of the ground state, a(0,0,0) (e 4/ )e c/6, contributes to Z AB with c the conformal anomaly of the theory. One thus obtains Z AB e c/6 A00B, /, 43 with 0Q0, j s 0, j f 0. The free energy is thus F AB ck B T 2 /6k B TlnA00B. 44 The second term in Eq. 44 is independent of the size of the system, and we therefore identify S imp k B lna00b 45 as the impurity contribution to the zero-temperature entropy. With no impurity present, AB and S imp 0, and it follows that A 0 consistent with the Fermi liquid gluing condition in Eqs. 21. Thus, from Eqs. 42 and 45 we finally get S imp k B ln2, 46 in agreement with the Bethe-ansatz result in Ref. 17. A few remarks may here be appropriate. First, we note that the effective renormalization of the charge sector, Q Qn c cf. Eq. 22, does not influence the impurity entropy. At first glance, this may seem obvious: the degeneracies of the impurity levels and hence the impurity entropy is a priori not expected to be affected by the potential scattering in Eq. 23 caused by the impurity valence n c. The picture becomes less trivial when one realizes that the impurity valence in fact controls how the spin and flavor degrees of freedom get screened by the electrons. As found in the Bethe-ansatz solution, 17 for nonzero s q, the quenching of the entropy occurs in two stages as the temperature is lowered. For sufficiently large values of, the free impurity entropy k B ln4 is first reduced to k B ln2 at an intermediate temperature scale, suggestive of a single remnant magnetic or quadrupolar moment depending on the sign of or, equivalently, the sign of n c 1/2). As the temperature is lowered further, this moment gets overscreened by the conduction electrons, as signaled by the residual entropy k B ln2. Remarkably, as found in Ref. 17 and verified here within the BCFT formalism, the same residual impurity entropy k B ln2 appears also in the mixed-valence regime PHYSICAL REVIEW B 68, where n c 1/2), for which there is no single localized moment present at any temperature scale. B. Impurity specific heat We now turn to the calculation of the impurity specific heat. Again, the analysis closely parallels that for the twochannel Kondo model. 19 Some elements are new, however, and we here try to provide a self-contained treatment. First, we need to write down the scaling Hamiltonian H scaling that governs the critical behavior close to the line of boundary fixed points. To leading order it is obtained by adding the dominant boundary operators found in Sec. III to the bulk critical Hamiltonian H* representing H bulk in Eq. 1: H scaling H* c J0 s O (s) 0 f O ( f ) 0 subleading terms, 47 with c,s, f the corresponding conjugate scaling fields. As reflected in the finite-size spectrum 22, H* splinters into dynamically independent charge, spin, and flavor pieces, implying, in particular, that all correlation functions decompose into products of independent charge, spin, and flavor factors. On the critical line the impurity terms H ion and H hybr in Eqs. 3 and 4 together masquerade as a boundary condition on H*, coded in the nontrivial gluing conditions in Eqs. 20. In addition H ion and H hybr also give rise to the exactly marginal charge current term c J(0) in Eq. 47. This term keeps track of the charge valence acquired by the impurity. As we have already noticed, this is different from the Kondo problem where the local spin exchange interaction at criticality gets completely disguised as a conformal boundary condition. 20 The reason for the difference is that away from the integer valence Kondo limit, the effective local scattering potential in Eq. 23 cannot be removed by redefining the charge quantum numbers Q Q2 in Eqs. 20 and 22. This in turn reflects the fact that for noninteger valence the dynamics manifestly breaks particle-hole symmetry, whereas for integer valence there is an equivalent description 17 in terms of the two-channel Kondo model where this symmetry is restored. 38 Off the critical line the impurity terms in Eqs. 3 and 4 appear in the guise of irrelevant boundary terms, of which s O (s) (0) and f O ( f ) (0) in Eq. 47 are the leading ones. Recall from Sec. II that a boundary critical theory by default is defined in the complex half-plane C Im z 0, with the boundary condition imposed on the real axis Im z0. As we have seen, by continuing the fields to the lower half-plane we obtain a chiral representation of this theory, now defined in the full complex plane C. Via the transformation wix(/) arctan(z), C can be mapped conformally onto an infinite cylinder, c call it, of circumference : c (,x);/2/2;x. It is convenient to use c as a finite-temperature geometry by treating as an inverse temperature. The partition function can then be written as